5.6 General Form of a Straight Line

The linear equation (first degree polynomial in two variables x and y ) ax by = (where a, b and c are real numbers such that at least one of a, b is non-zero) always represents a straight line. This is the general form of a straight line. Now, let us find out the equations of a straight line in the following cases (i) parallel to ax + by + c = (ii) perpendicular to ax + by + c = . .

Equation of a line parallel to the line ax by = The equation of all lines parallel to the line ax by = can be put in the form ax by k = for different values of k . . . Equation of a line perpendicular to the line ax by = The equation of all lines perpendicular to the line ax by = can be written as bx ay k = for different values of k .

Coordinate Geometry Two straight lines a x b y and a x b y where the coefficients are non-zero, are (i) parallel if and only if a ; That is, a b a b (ii) perpendicular if and only if a a b b Progress Check Fill the details in respective boxes No. Equations Parallel or perpendicular S.No. Equations Parallel or perpendicular x – y + = . .

Slope of a straight line The general form of the equation of a straight line is ax by = . (at least one of a , b is non-zero) coefficient of x , coefficient of y , constant term = c . The above equation can be rewritten as by = − ax gives y = − b x b , if b ¹ … ( ) comparing ( ) with the form y mx l We get, slope m = − a m = − coefficientof coefficientof y intercept l = − c y intercept = − constantterm coefficientof y Example . Find the slope of the straight line Solution Given slope m = − coefficientof coefficientof y = − = − Therefore, the slope of the straight line is - .

Example . Find the slope of the line which is (i) parallel to (ii) perpendicular to Thinking Corner How many straight lines do you have with slope ? Solution (i) Given straight line is ⇒ – = Slope m = − Since parallel lines have same slopes, slope of any line parallel to is . (ii) Given straight line is Slope m = − = Since product of slopes is − for perpendicular lines, slope of any line perpendicular to is − = − Example .

Show that the straight lines and are parallel. Solution Slope of the straight line is m = − coefficient of coefficient of m = − Slope of the straight line is m = − = − Here, m = m That is, slopes are equal. Hence, the two straight lines are parallel. Example .

Show that the straight lines x and are perpendicular. Solution Slope of the straight line x is m = − Slope of the straight line is m = − = − Now, m m × − = − Hence, the two straight lines are perpendicular. Example . Find the equation of a straight line which is parallel to the line and passing through the point ( , ) .

Aliter a , b a , b Therefore, a Hence the lines are parallel. Aliter a = , b = − ; a , b a a b b The lines are perpendicular. Coordinate Geometry Solution Equation of the straight line, parallel to is k Since it passes through the point ( , ) + k = k = = Therefore, equation of the required straight line is Example . Find the equation of a straight line perpendicular to the line y and passing through the point ( , – ) .

Solution The equation y can be written as Equation of a straight line perpendicular to is k Since it is passes through the point ( , – ) , + k = we get, k = − Therefore, equation of the required straight line is Example . Find the equation of a straight line parallel to Y axis and passing through the point of intersection of the lines and x .